Question 244092
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First thing:  You cannot find (or write) "the" equation of a line.  There are an infinity of equivalent representations of any line, hence the best you can do is find "an" equation of a line.


If you are given a point and the slope, then use the point-slope form of a line:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y - y_1 = m(x - x_1) ]


Where *[tex \LARGE m] is the given slope and *[tex \Large \left(x_1,y_1\right)] is the given point.


If you are not given the slope directly, but are given that the desired line is either parallel or perpendicular to a given line, use one of the following to determine the slope:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  L_1 \parallel L_2 \ \ \Leftrightarrow\ \ m_1 = m_2]



*[tex \LARGE \ \ \ \ \ \ \ \ \ \  L_1 \perp L_2 \ \ \Leftrightarrow\ \ m_1 = -\frac{1}{m_2} \text{ and } m_1, m_2 \neq 0]


If you are given two points, you can either compute the slope using the slope formula,


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m = \frac{y_1 - y_2}{x_1 - x_2} ]


Where *[tex \Large \left(x_1,y_1\right)] and *[tex \Large \left(x_2,y_2\right)] are the coordinates of the given points


And then use the point-slope form as above, or you can use the two-point form of a line:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y - y_1 = \left(\frac{y_1 - y_2}{x_1 - x_2}\right)(x - x_1) ]


Which is really nothing more than a combination of computing the slope and writing the equation.


In all cases, you should determine, either from the instructions in your text or from your instructor's guidance, what form your answers should take.  Generally, it is one of two forms that are specified, if any:


1.  Slope-intercept form:  *[tex \LARGE y\ =\ mx\ +\ b] where *[tex \LARGE m] is the slope and *[tex \LARGE \left(0,b\right)] is the point of intersection of the line with the *[tex \LARGE y]-axis.


2.  Standard form:  *[tex \LARGE Ax\ +\ By\ =\ C]  Some text require *[tex \LARGE A], *[tex \LARGE B], and *[tex \LARGE C] to be integers for proper standard form.


A circle centered at *[tex \Large \left(h,k\right)] and with radius *[tex \Large r] is described by:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (x - h)^2 + (y - k)^2 = r^2]


Since you were given the center and a point on the circle, you will have to calculate the radius by using the distance formula to determine the distance from the center to the given point.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ r\ =\ d\ =\ \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}]


Again where *[tex \Large \left(x_1,y_1\right)] and *[tex \Large \left(x_2,y_2\right)] are the coordinates of the given points.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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